
for almost all \(t\in [0,T]\) .
We thus obtain that \(\big (f(t),g(t)\big ) \in \mathcal {H}^1\) 
and by regularity theory for elliptic problems with bulk-surface coupling (see [1]}), we infer that \(\big (\phi (t),\psi (t)\big ) \in \mathcal {H}^3\)  with
\(&\big \Vert  \big (\phi (t),\psi (t)\big ) \big \Vert _{\mathcal {H}^3}^2\le C \big \Vert  \big (f(t),g(t)\big ) \big \Vert _{\mathcal {H}^1}^2\\&\quad \le C \Big ( \Vert  \mu (t) \Vert _{H^1(\Omega )}^2+ \big \Vert  F^{\prime }\big (\phi (t)\big ) \big \Vert _{H^1(\Omega )}^2+ \Vert  \theta (t) \Vert _{H^1(\Gamma )}^2+ \big \Vert  G^{\prime }\big (\psi (t)\big ) \big \Vert _{H^1(\Gamma )}^2 \Big )\\&\quad \le C +C\Vert  \mu (t) \Vert _{H^1(\Omega )}^2+ C \Vert  \theta (t) \Vert _{H^1(\Gamma )}^2 + C\epsilon ^2 \big \Vert  \big (\phi (t),\psi (t)\big ) \big \Vert _{\mathcal {H}^3}^2\) 
